Quantum error correction (QEC) is the set of techniques used to protect quantum information from errors caused by decoherence and imperfect gates. Unlike classical bits, which can be copied and checked against redundant copies, quantum states cannot be cloned — any measurement collapses the state. QEC works around this by encoding a single logical qubit into an entangled state of several physical qubits, allowing errors to be detected and corrected without learning anything about the encoded information.

Noise in a quantum system can be modelled as interaction with an environment: a nominally isolated qubit leaks phase or energy into its surroundings, turning a pure state into a mixed state described by a density matrix $\rho$ rather than a state vector. The time evolution of $\rho$ under both coherent Hamiltonian dynamics and incoherent dissipation is captured by the Lindblad equation. The same evolution can be expressed in the Kraus operator-sum representation $\mathcal{E}(\rho) = \sum_k K_k \rho K_k^\dagger$, which treats the environment implicitly through the operators $\{K_k\}$.

Any single-qubit error can be decomposed as a linear combination of the four Pauli operators $\{I, X, Y, Z\}$, so it suffices to detect and correct only bit-flip ($X$), phase-flip ($Z$), and combined ($Y$) errors. The quantum error correction codes that do this rely on measuring error syndromes — multi-qubit observables that reveal which error occurred without revealing the logical state. The threshold theorem states that if the physical error rate is below a code-dependent threshold, arbitrarily long computations are possible by concatenating or tiling error-correcting codes.